Compressive Sensing
Compressive sensing (CS) has redefined signal acquisition systems, as well as signal processing systems. Conventional systems rely on a fine, high-rate sampling, intending to acquire as much information as possible before processing the signal. In contrast, CS uses an appropriate signal model, such as sparsity, so that the burden of frequent sampling can be significantly reduced, perhaps at the expense of increased processing that incorporates the signal model.
Array Signal Processing
Array signal processing is a field always in search of ways to reduce sampling complexity. Scenes sensed by array processing, at least when background and clutter is suppressed, are often very sparse in the spatial domain, or in the case of imaging systems, after an appropriate basis transformation. For those reasons, the array processing field has used CS both in array design and in processing methodology.
A typical sensing array in a receiver includes a number of sensor elements, each sensing a signal field. During passive sensing, the signal field is generated by transmitters (sources) in a scene, and a shape and timing of transmitted signals is not known. During active sensing, the array transmits, and then receives signals (echoes) reflected by the scene. In either case, the goals is to reconstruct the scene based, as best as possible, from the received signal, to determine, for example, whether objects in the scene are transmitters or reflectors, in the case of the passive array, or reflectors in the case of the active array.
Compressive Sensing and Sampling
CS systems measure signals of interest at a rate determined by an information content of the signal, typically measured by the number of non-zero components, which characterizes sparsity if the number is substantially small compared to the number of zero components.
Typically, the measuring is according to a linear acquisition systemr=As+n, 
where r denotes the sensed data, s a sensed signal, n acquisition noise, and A a matrix describing the linear acquisition system.
The signal is assumed to be K-sparse or K-compressible in some basis, i.e., the K largest components of the signal. This basis acquires all or most of the energy of the signal. Under this assumption, the signal can be recovered by solving a convex optimization problem
      s    ^    =            arg      ⁢                          ⁢                        min          s                ⁢                                                          s                                      1                    ⁢                                          ⁢                      s            .            t            .                                                  ⁢            r                                ≈          A      ⁢                          ⁢      s      or a greedy procedure minimizing
      s    ^    =            arg      ⁢                          ⁢                        min          s                ⁢                                                                          r                -                                  A                  ⁢                                                                          ⁢                  s                                                                    2            2                    ⁢                                          ⁢                      s            .            t            .                                                  ⁢                                                          s                                            0                                            ≤          K      .      
Under certain conditions on the matrix A, those methods are guaranteed to provide the correct solution even with noise and model mismatches.
Restricted Isometry Property (RIP)
A restricted isometry property (RIP) characterizes matrices that are nearly orthonormal when operating on sparse vectors. RIP one of the best known and widely used condition to provide the above guarantees. The matrix A satisfies the RIP of order A with an RIP constant δ if(1−δ)∥s∥22≦∥As∥22≦(1+δ)∥s∥22,for all K-sparse signals s. In other words, the linear system satisfies the RIP if it preserves the norm of K-sparse signals. To guarantee recovery of the K-sparse signals and accuracy, the system must satisfy the RIP of order 2K with constant δ≦√{square root over (2)}−1.
Since the advent of CS, a large number of practical ways of implementing systems exhibiting the RIP have been developed. The systems of interest are based on time-domain systems, such as a random subsampler or a random demodulator, which are designed to measure time signals sparse in the frequency domain. All of those systems implement a linear acquisition systemr=AF−1s,  (1)where F is a discrete Fourier transform (DFT) matrix, and s is the sparse or compressible frequency representation of the time domain signal x=F−1s. The goal of those designs is to ensure the system AF−1 satisfies the RIP.